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Modeling Financial Swaps and Geophysical Data Using The View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NDSU Libraries Institutional Repository MODELING FINANCIAL SWAPS AND GEOPHYSICAL DATA USING THE BARNDORFF-NIELSEN AND SHEPHARD MODEL A Dissertation Submitted to the Graduate Faculty of the North Dakota State University of Agriculture and Applied Science By Semere Kidane Habtemicael In Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Department: Mathematics November 2015 Fargo, North Dakota NORTH DAKOTA STATE UNIVERSITY Graduate School Title MODELING FINANCIAL SWAPS AND GEOPHYSICAL DATA USING THE BARNDORFF-NIELSEN AND SHEPHARD MODEL By Semere Kidane Habtemicael The supervisory committee certifies that this dissertation complies with North Dakota State Uni- versity's regulations and meets the accepted standards for the degree of DOCTOR OF PHILOSOPHY SUPERVISORY COMMITTEE: Indranil SenGupta Chair Azer Akhmedov Nikita Barabanov Sylvio May Approved: November 17, 2015 Benton Duncan Date Department Chair ABSTRACT This dissertation uses Barndoff-Nielsen and Shephard (BN-S) models to model swap, a type of financial derivative, and analyze geophysical data for estimation of major earthquakes. From empirical observation of the stock market activity and earthquake occurrence, we observe that the distributions have high kurtosis and right skewness. Consequently, such data cannot be captured by stochastic models driven by a Wiener process. Non-Gaussian processes of Ornstein-Uhlenbeck type are one of the most significant candidates for being the building blocks of models of financial economics. These models offer the possibility of capturing important distributional deviations from Gaussianity and thus these are more practical models of dependence structures. Introduced by Barndorff-Nielsen and Shephard these processes are not only convenient to model volatility in financial market, but have an independent interest for modeling stationary time series of various kinds. For the financial data we use BN-S models to price the arbitrage-free value of volatility, variance, covariance, and correlation swap. Such swaps are used in financial markets for volatility hedging and speculation. We use the S&P 500 and NASDAQ index for parameter estimation and numerical analysis. We show that with this model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. For any given time interval, the earthquake magnitude data have three main properties: (1) magnitude is a non-negative stationary stochastic process; (2) for any finite interval of time there are only finite number of jumps; (3) the sample path of the magnitude of an earthquake consists of upward jumps (significant earthquake) and a gradual decrease (aftershocks). For such geophysical data we specifically use Gamma Ornstein Uhlenbeck processes driven by a L´evyprocess to estimate a major earthquake in a certain region in California. Rigorous regression analysis is provided, and based on that, first-passage times are computed for different sets of data. Both applications demonstrate the significance of BN-S models to phenomena that follow non-Gaussian distributions. iii ACKNOWLEDGEMENTS First I would like to thank ALMIGHTY GOD for giving me the audacity and sanctioning me with the confidence to fulfill this daunting task. I am indubitably grateful to my Committee Chairman, Dr. Indranil SenGupta, for his excellent guidance, caring, and patience; for providing me with an excellent atmosphere for completing this dissertation; and for soliciting additional resources throughout my Doctoral Program endeavor. I benefitted from his provocative thoughts and discussions. He provided his perceptive comments and/or constructive feedback in near real- time. Although it is impossible to list everyone to whom I am indebted, several persons deserve special mention. My mentor Drs. Indranil SenGupta, Azer Akhmedov, Nikita Barabanov, and Sylvio for accepting to serve and their support. Without all their kind assistance, continuous support, and insightful feedback, I would not achieve and accomplish what seems to be too far to reach in a lifetime. I would like to extend a note of thanks to Mr. Curt Doetkott, Instructor of SAS/Lead of the Statistical Consulting Services unit, at NDSU, for his untiring assistance while writing my dissertation. My deepest appreciation and a profound thankful note also goes to Dr. Muise Ghe- bremichael for his strenuous effort, motivation, sage guidance, who took the initiative to shepherd me throughout this tortuous academic journey. Dr. Ghebremichael has a special memories in my personal life, and no words could adequately express my gratitude. My wholehearted and sincere gratitude/appreciation also goes to Dr. Semhar Michael, her husband Dr. Johannes Afework and Mr. Noah Lebasi for their hospitality, knowledge, and wisdom have supported and enlightened me over many years of friendship. Their altruistic support have been invaluable during my doctoral program and while writing this dissertation. A note of thanks also goes to our Secretary Ms. Milan Melanie, Department of Mathematics NDSU, for all her friendliness, assistance and support throughout my tenure at NDSU. I thank the faculty and graduate assistants of the Mathematics department, North Dakota state University, for their advice and reminders over the last three and half years. Special thank goes to Melissa Duchsherer, John Laumann-Belth, and Matt Warner(graduate writers center), for accepting to proofreading my dissertation. A thankfull note also goes to the graduate students of iv department of statistics, learning statistics class was fun with you guys all. I had the apportunity to learn what seems impossible in my career. I would like to extend special thanks to Fesseha Gebremikael, a Ph. D. student at Upper Great Plains Transportation Institute (UGPTI) for his countless befitting brotherly suggestions, encouragement, who took the liberty to review, and edit my dissertation. Discussions with Fesseha always left me empowered and invigorated, and for all those moments I am very thankful. My thanks also goes to my Fargo friends, Rufael, kefali with his family, Mahder, Tadele, and to my colleague Melisa and Cory, who assisted, advised, and supported my efforts over the years. I would like also to thank friends from Grand Forks area, Gashaw, Tesfay, Dr. Chernet, Dr. Daba, Enque, and Dr. Belete. My extraordinary gratitude is also extended to my dearest parents my father Mr. Kidane Gebreselassie my mother Wezero Tsehaynesh Unquay have been the the primary source of moti- vation and support for education and career. You have educated me, you have encouraged me to follow my dreams, you have been the best parents one could wish for. I thank you from the bottom of my heart. I thank my brothers Mebrahtu (with his family), Dawit, Mehari, Robiel, Kibrom, and Aron andmy sisters Milasu (with her family),Nazret, and Elsu for the love and continued support. I am forever indebted to my parents for their understanding, endless patience and encouragement when it was most required. Throughout the writing of this dissertation, I have had the love, full support and tenacity of my wife Akiberet (Akutey) and my lovely God's gift son Nathan. They are the love of my life, the rock that I can lean on and the ones who shares my best and worst moments. I greatly appreciate each one of them, for their compassionate encouragement throughout this journey. Without them, this proud and victorious episode would not have been achievable. v DEDICATION I would like to dedicate this doctoral dissertation to my loving parents, brothers and sisters, and of course, my wife and son. Thanks so much for their never-ending love, support, and spirit, during this tedious period. vi TABLE OF CONTENTS ABSTRACT . iii ACKNOWLEDGEMENTS . iv DEDICATION . vi LIST OF TABLES . ix LIST OF FIGURES . x 1. INTRODUCTION . 1 1.1. Types of Volatility . 4 1.1.1. Historical Volatility . 4 1.1.2. Implied Volatility . 4 1.2. Recent Developments in Swaps . 5 2. THE HULL -WHITE MODEL FOR DERIVING SWAP . 10 2.1. Introduction . 10 2.2. Variance and Volatility Swap . 11 2.2.1. Variance Swap . 11 2.2.2. Volatility Swap . 12 2.3. Covariance and Correlation Swap . 15 2.3.1. Covariance Swap . 15 2.3.2. Correlation Swap . 18 3. BARNDORFF-NIELSEN AND SHEPHARD MODEL FOR STOCK AND VOLATILITY DYNAMICS . 20 4. PRICING VARIANCE AND VOLATILITY SWAP FOR BN-S MODEL . 30 4.1. Variance Swap . 30 4.2. Volatility Swap . 31 4.3. Closed Form Solution of Volatility Swap . 31 4.4. Model Fitting and Parameter Estimate . 39 vii 4.5. Conclusion . 43 5. PRICING COVARIANCE AND CORRELATION SWAP FOR TWO ASSETS WITH A STOCHASTIC VOLATILITY . 47 5.1. Pricing Covariance Swap . 47 5.2. Pricing Correlation Swap . 57 5.3. Model Fitting and Parameter Estimate . 61 5.4. Conclusion . 62 6. ORNSTEIN-UHLENBECK PROCESS FOR GEOPHYSICAL DATA ANALYSIS . 66 6.1. Introduction . 66 6.2. Ornstein-Uhlenbeck Processes for Geophysical Modeling . 69 6.3. Computation of Characteristic Function . 72 6.4. Analysis of the First-Passage Time . 74 6.5. Statistical Analysis of the Data . 77 6.5.1. Geophysical (Earthquake) Data . 78 6.5.2. Regression Analysis and Estimation of Major Events . 78 6.6. Conclusion . 80 REFERENCES . 85 viii LIST OF TABLES Table Page 4.1. Parameter estimate of Gamma distribution for variance swap . 40 4.2. Parameter estimate of Inverse Gaussian for variance swap . 40 4.3. Parameter estimate of Positive tempered stable for variance swap . 41 4.4. Comparing errors of different models for variance swap . 41 4.5. Parameter estimates for Γ volatility swap with Theorem 4.2.2 . 43 4.6. Parameter estimates for IG volatility swap with Theorem 4.2.2 . 43 4.7. Parameter estimates for PTS volatility swap with Theorem 4.2.2 . 44 4.8. Comparing errors of different models for volatility swap . 44 4.9. Parameter estimates for Γ volatility swap with Theorem 4.3.3 . 45 4.10. Parameter estimates for IG volatility swap with Theorem 4.3.3 .
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